5-3 Continued (Adding & Subtracting Mixed Numbers)

Fractions are pesky, aren't they? They're always asking us to change them around.

Read through the rest of p. 248 and 249 as a class.

Let's sum up what we learned. Sometimes, when adding and subtracting with mixed numbers, I have to think of myself as a secret, government-appointed scientist who has to solve an important calculation before an asteroid destroys the earth, just 'cause fractions irritate me that much. But - if you use the super-secret fraction-action method, you'll be fine.

1) Convert your mixed numbers into improper fractions. Here's a diagram from mrmelsonmath624.blogspot.com to remind you how we do this:


If you don't remember how to convert mixed numbers to improper fractions, you'll need to practice that before you move on. Since I can't accurately display mixed numbers from this keyboard, I'd like you to look at pg. 250, #15.

Once you convert those mixed numbers to improper fractions, you should have:
81/8 + 15/4.

2) Now we proceed as with regular fractions. Since we can't add them 'til they've got the same bottom number, we find the LCM of the denominators. The LCM of 8 and 4 is 8.

3) Convert the fractions to new fractions that have 8 as the denominator (thankfully, we only need to convert one of them):

15/4 = ?/8    ---> We know we can multiply 4 by 2 to get 8, so we'll do the same thing to the top:

15/4 = 30/8

4) Now, add your new fractions: 81/8 + 30/8 = 111/30.

5) Finally, simplify. This will most likely result in a mixed number. Remember, to convert (or simplify) an improper fraction to a mixed number, we divide the top by the bottom, and use the remainder as the top of the fraction and the original denominator for the bottom. Here's a diagram from ricksmath.com:



In our case, we divide: 111 ÷ 30.

111 ÷ 30 = 3 remainder 21. So, for our mixed number, our whole number is 3, the top of our fraction is 21, and the bottom is 30:

3 and 21/30 (again, sorry I can't display mixed numbers properly on here). But wait! Our mission is not done. Our answer is not in simplified form. Our whole number stays the same, but the fraction can be simplified. Our final answer is:

3 and 7/10.

I know this is a lot to take in. It takes practice and lots of remembering what you've already learned about fractions, but you can do it!

Are you ready to try p. 250, 15 - 18?

I'll be posting more tomorrow on 5-4. It'll be there when you're ready. Also, please take a look around the room - there's a fractions poster on the Math board that will remind you of some fraction principles.

Adding and Subtracting Fractions

The very first thing I'd like you to do for this section is to read through 5-3 in your textbook and STOP at Section 2 (Adding and subtracting with mixed numbers). Don't worry about notes or quickchecks or anything - just simply read this part of 5-3 silently to yourself.

Now, let's sum everything up:

Fractions can be added and subtracted as long as they have the same denominator (bottom number). Kind of like being in the same "family" - everyone has the same last name.

In the first section, we learned that we can easily add and subtract fractions with the same denominator simply by adding or subtracting the numbers on top. Basically, solve the problem with the TOP numbers and leave the bottom number the same (and don't forget to simplify if necessary!)

Example:
 3/5 and 1/5 have the same denominator, so I can easily add them by just adding the top numbers and keeping the denominator the same:

3/5 + 1/5 = 4/5.

The tricky part comes in when you have two fractions with different denominators. Before we can add or subtract them, we have to give them the same denominator (put them into the same "family"). So, how do we do this?

1) Find the LCM of the two denominators. The example in the book shows us 2/3 + 1/5. Since our denominators here are 3 and 5, our LCM = 15.

2) Now we need to change our fractions to new fractions that have a denominator of 15. Recall that to find any equivalent fraction, we must do the same thing to the top that we did to the bottom. So, 2/3 equals SOMETHING over 15, and 1/5 also equals SOMETHING OVER 15:

2/3 = ?/15
1/5 = ?/15

To find out what goes where the question marks are, figure out how we got from the old denominators to the new denominators. For the first one (2/3), how did we get to 15? We know that we can multiply 3 by 5 to get 15. Now, do the same thing to the top: 2 x 5 = 10. Our new fraction is 10/15.

Let's figure out the second fraction (1/5). How did we get from a denominator of 5 to a denominator of 15? we know we can multiply 5 by 3 to get 15. Now, do the same thing to the top: 1 x 3 = 3. Our new fraction is 3/15.

3) Now our new fractions can finally be added because they have the same denominator!

10/15 + 3/15 = 13/15. ---> This is already in simplest form, and so it is our answer.

To sum up the summary (lol!):

- Fractions need to have the same denominator in order to be added or subtracted.
- If they already have the same denominator, just add or subtract the top numbers, leave the bottom the same, and simplify if necessary.
- If they do NOT have the same denominator:
1) Find the LCM of the denominators.
2) Convert the fractions to equivalent fractions with the LCM as your common denominator.
3) Solve.

Try p.250, 5 - 12 individually. Check as a class. When you're ready to continue, the next post will address the rest of 5-3.

P.S.: To L-Dubs - thank you SO much for my Lyme Disease plushie! It's seriously the cutest thing ever. Made me smile. His name is Lymie. I will post pics of Lymie helping me work soon. :)

Review and Repeating Decimals

Hi guys. Let's review what we already know about converting decimals to fractions. Remember this step-by-step guide when you get stuck.

1) NAME the decimal (use your place value chart to find out what it "says").
    Example: .075 would be "seventy-five hundredths" since the last digit ends in the hundredths place
                   .6 would be "six tenths" because the number ends in the tenths place
                   1.457 would be "one and four hundred fifty-seven thousandths" because the last digit ends in the
                             thousandths place

2) Write your fraction. Let's use .075 as our example. Since we know .075 is "seventy-five hundredths", we write our fraction out as 75/100.

3) Reduce to simplest terms. 75/100 simplified would be 3/4 (remember, this is where we use the GCF to simplify)

Once you're able to turn decimals into fractions, all we need to do is use a little algebra to figure out how to turn those pesky repeating decimals into fractions. I'm not a big fan of the way your textbook explains it, so let me see if I can explain it in a simpler way.

Let's take the following number as an example (the same one in your textbook) - remember to click on the pictures to make them bigger:

1) Step one: Ignore the repeating symbol (pretend it's not even there). Move the decimal to the END of the number until it's a whole number.

2) Count how many times you moved your decimal to the right. In our case, it was TWO times.

3) Raise the number 10 to THAT power (the number of times you moved the decimal). In our case, we moved it TWO times, so we raise 10 to the second power:

4) Here comes the weird part. Subtract 1 from whatever number you got when you did step 3. In our case, we got 100. 100 - 1 = 99.

5) Now set up an equation that says "99x equals 72" (we are bringing back that "72" from when we moved the decimal in step 1).

6) Use what you already know about algebraic equations to solve for "x" by dividing both sides by 99.

7) We now have a fraction on the right side. Use the GCF (in this case, 9) to divide down and simplify your fraction.

So our answer is 8/11.



Here are the first two quick checks for this concept written out using our steps (sorry the second one is a little blurry):

I know it's tough but if you keep going through these steps and examples I know you'll get the hang of it. Try the third Quick Check as a class and let me know how it goes!

Monday, 4/15

Happy Monday everyone. Please check your answers to Friday's assignment (pg. 244, 1 - 12).

Please let me know if you guys had any trouble with these. If not, today you can  move on to more of 5-2. Read through the first half of pg. 243 as a class. You'll be learning how to turn a decimal into a fraction. Check back here once you've read through the first example (example 4).

In order to change a decimal to a fraction, you must first think back to when you learned about place value and be able to say what a decimal is correctly. Here is a place value chart from mathtutorvista.com to remind you:



Let's do the first Quick Check together.

1) 1.75 --- We would call this "one and seventy-five hundredths"
     Now we write the number out as a fraction using the fraction we just said:
     1 75/100 (sorry, I can't write fractions the normal way here on the blog)
     --- Since 75/100 is not in simplest form, simplify. Our new number is:
     1 3/4 (One and three fourths)

Here's another example:

Rewrite .6 as a fraction.

--- We would read .6 as "six tenths" - which gives us the fraction 6/10. Now simplify. We can write our
      fraction as 3/5.



We aren't going to move on to repeating decimals until tomorrow. For today, practice what you've just learned. Complete pg. 244, 16 - 27.

Friday, 11/12

Happy Frrrrriiiidaaayyyyyyyyyyyy!!!!!

Here are the answers to yesterday's questions (p. 760, 1 - 18):

1) 30
2) 40
3) 24xy
4) 15t²
5) >
6) < 
7) = 
8) > (Remember, we're dealing with negatives for this one)

For today's class, I'd like you to read through pgs. 241 and 242 as a class. Please try the quick checks along the way on your laptop after you've read each Example. You'll see through the first example that, in order to turn a fraction into a decimal, just divide the top by the bottom.*Remember that any whole number can be written with a decimal. In the first example, to turn 5/8 into a decimal, we had to divide 5 by 8. In order to do so, rewrite "5" as "5.0", carry the decimal up, and then begin to divide. 

We are still not yet using calculators for Chapter 5.

The second example shows how to indicate that you have a repeating decimal. For example, if you divide a fraction to transform it into a decimal and you find yourself continuing to divide - over and over again, getting the same number - that's called a "repeating" decimal and can be indicated with a bar over the numbers that repeat. 

The third example will show you that in order to compare a mixed group of fractions and decimals, turn each fraction into a decimal and then compare. 

Once you have mastered the Quick Checks, you can move on to your Classwork Assignment:
pg. 244, 1 - 12 

(If you don't finish in class, please finish for Homework)

Have a fantastic weekend!

Thursday, 4/11

Hello everyone! I hope you are all enjoying this crazy awesome weather. If the font for these posts is too small to see from your seat, leave a comment and I'll change the font size for next time (or your can use the "zoom" tool back on my computer).

For today's lesson I'd like to review what we covered in 5-1.

First, we learned how to find the Least Common Multiple (LCM) of two numbers, either using the listing method or using prime factorization. Let's review the prime factorization method:

First, find the prime factorization of the two numbers:

               8                   18
           2 x 4               2 x 9
        2  x 2 x 2         2 x 3 x 3

Our prime factorizations are at the bottom. Now, we need to multiply all of the bottom "tree" numbers together, using the matches ONLY ONCE. Our only match between the two tree bottoms is a pair of twos (in red), so we use it only once, and then bring all the other numbers down to multiply:

2 x 2 x 2 x 3 x 3 = 72

The LCM of 8 and 18 is 72.

Let's try it with three numbers. Find the LCM of 6, 14, and 28 using prime factorization:

First, find the prime factorization of all three numbers (I'm going to skip the whole tree and go straight to the answers we'd have at the bottom):

      6                       14                      28
   2 x 3                   2 x 7               2 x 2 x 7      

We have a matching 2 for all three trees, so we use it once. We also have a match of 7 between 14 and 28, so we use it only once. Then, we multiply those with all of the leftover, non-matching numbers:

2 x 7 x 3 x 2 = 84. The LCM of 6, 14, and 28 is 84.

Okay, everyone take a two-minute brain break by using Squidward hands to greet the people next to you and ask them how their day has been.

Now. What on earth does all this LCM stuff have to do with fractions? Remember, in ancient times (a couple of days ago) when we had to compare fractions, that we used the LCM to turn fractions with different denominators into fractions with the same denominators. (Because you can't compare two fractions that have different denominators.) Kinda like giving them the same last name, or putting them into the same "family."

In order to compare fractions with unlike denominators, you must:

1) Find the LCM of the denominators (now called the Least Common Denominator).
2) Convert the original fractions into equivalent fractions with the Least Common Denominator.
3) Compare the numerators, and voila!

Let's walk through an example using these steps:

Which is greater, 2/3 or 7/8? (Sorry about the sideways fractions!)

1) Find the LCM of the denominators:
          3                   8
          3               2 x 2 x 2 
We have no matches, so we multiply everything together: 3 x 2 x 2 x 2 = 24. This is now the Least Common Denominator we're looking for. 

2) Convert the original fractions into equivalent fractions with the Least Common Denominator.
That means we need to change 2/3 and 7/8 into fractions that have 24 at the bottom.

2/3 = ?/24 --- First, figure out how we can get from 3 to 24. Once we realize we multiply by 8, we must do the same thing to the numerator. Our new fraction is 16/24.

7/8 = ?/24 --- How do we get from 8 to 24? We multiply by 3. Do the same thing to the top for our new fraction: 21/24.

3) Now we can finally compare the two fractions by simply looking at the numerator. Which is bigger, 16/24 or 21/24?
We know that 21/24 is bigger. Therefore, 7/8 is greater than 2/3. 

And now, Mathlings, we are fully equipped to compare fractions and to list them in order from least to greatest, even if they have different denominators. Let's do one more exercise to practice all this stuff:

Classwork: pg. 760, 1 - 8. Please use the prime factorization method for finding the LCM. We will check answers tomorrow (or, if everyone finishes in class, check answers together with Mrs. Croskey).

Roscoe says, "Have a superawesometerrific Thursday!"

Wednesday 4/10

Greetings, Mathlings. Since you have a short class today because of testing,  take the opportunity to share some things about yourself with Mrs. Croskey! I suggest a round of the favorites game, Have You Ever, and maybe even a little silly story time. Tomorrow we get back into Chapter 5. Have fun!!!